Monte Carlo Treatment Planning

An Introduction

NEDERLANDSE COMMISSIE VOOR STRALINGSDOSIMETRIE

Report 16 of the Netherlands Commission on Radiation Dosimetry

Netherlands Commission on Radiation Dosimetry

Subcommission Monte Carlo Treatment Planning

June 2006

Monte Carlo Treatment Planning

An Introduction

NEDERLANDSE COMMISSIE VOOR STRALINGSDOSIMETRIE

Report 16 of the Netherlands Commission on Radiation Dosimetry

Authors:

N. Reynaert

S. van der Marck

D. Schaart

W. van der Zee

M. Tomsej

C. van Vliet- Vroegindeweij

J. Jansen

M. Coghe

C. De Wagter

B. Heijmen

Netherlands Commission on Radiation Dosimetry

Subcommission Monte Carlo Treatment Planning

June 2006

i

Preface

The Nederlandse Commissie voor Stralingsdosimetrie (NCS, Netherlands Commission on

Radiation Dosimetry) was officially established on 3 September 1982 with the aim of

promoting the appropriate use of dosimetry of ionizing radiation both for scientific research

and practical applications. The NCS is chaired by a board of scientists, installed upon the

suggestion of the supporting societies, including the Nederlandse Vereniging voor

Radiotherapie en Oncologie (Netherlands Society for Radiotherapy and Oncology), the

Nederlandse Vereniging voor Nucleaire Geneeskunde (Netherlands Society for Nuclear

Medicine), the Nederlandse Vereniging voor Klinische Fysica (Netherlands Society for

Clinical Physics), the Nederlandse Vereniging voor Radiobiologie (Netherlands Society for

Radiobiology), the Nederlandse Vereniging voor Stralingshygine (Netherlands Society for

Radiological Protection), the Nederlandse Vereniging voor Medische Beeldvorming en

Radiotherapy (Netherlands Society for Medical Imaging and Radiotherapy), the Nederlandse

Vereniging voor Radiologie (Netherlands Society for Radiology) and the Belgische

Vereniging voor Ziekenhuisfysici/Socit Belge des Physiciens des Hpitaux (Belgian

Hospital Physicists Association.

To pursue its aims, the NCS accomplishes the following tasks: participation in dosimetry

standardisation and promotion of dosimetry intercomparisons, drafting of dosimetry

protocols, collection and evaluation of physical data related to dosimetry. Furthermore the

commission shall maintain or establish links with national and international organisations

concerned with ionizing radiation and promulgate information on new developments in the

field of radiation dosimetry.

Current members of the board of the NCS:

S. Vynckier, chairman

B.J.M. Heijmen, vice-chairman

E. van Dijk, secretary

J. Zoetelief, treasurer

A.J.J. Bos

A.A. Lammertsma

J.M.Schut

F.W. Wittkmper

D. Zweers

ii

Monte Carlo Treatment Planning: An Introduction

This report was prepared by a subcommittee of the Netherlands Commission on Radiation

Dosimetry (NCS), consisting of Belgian and Dutch scientists.

Members of the subcommittee:

N. Reynaert, chairman

S. van der Marck

D. Schaart

W. van der Zee

M. Tomsej

C. Van Vliet-Vroegindeweij

J. Jansen

M. Coghe

C. De Wagter

B. Heijmen

Monte Carlo Treatment Planning: An Introduction

Report 16 of the Netherlands Commission on Radiation Dosimetry (NCS)

June 2006

NCS, Delft, The Netherlands

ISBN 90-78522-01-1

For more information on this and other NCS Reports, see http://www.ncs-dos.org

iii

User guide

This report presents an overview of the literature for physicists in radiotherapy departments

who intend to buy/use/customise a Monte Carlo treatment planning system for electron

and/or photon therapy. The report focuses on commissioning, selection of treatments

requiring Monte Carlo, variance reduction techniques, accelerator head modelling, patient

modelling (conversion of CT Hounsfield units), hardware requirements and the required

knowledge to operate an MCTP system. In addition an overview of existing Monte Carlo dose

engines and MCTP systems is given.

The report consists of three main parts.

The first part provides insight in the Monte Carlo method for dose calculations. An overview

of general purpose Monte Carlo codes, used in the field of electron and photon dosimetry, is

given. An extensive description of modelling of electron and photon transport and the usage

of cross sections is presented.

The second part deals with MCTP specific topics such as CT conversion, linac head

modelling, scoring, variance reduction, Monte Carlo based treatment planning (optimisation),

and 4D planning.

The third and final part focuses on practical aspects. It provides an overview of Monte Carlo

dose engines used for Monte Carlo treatment planning, an overview of commercial MCTP

systems, and guidelines on benchmarking of these systems (focussing on MC specific

benchmarks).

iv

Contents PREFACEI

USER GUIDE....III

CONTENTS..IV

SUMMARY ........................................................................................................................1

ABBREVIATIONS ..................................................................................................................3

1 INTRODUCTION.....................................................................................................5

PART I: INTRODUCTION TO MONTE CARLO......................................................................6

2 MONTE CARLO FOR SOLVING NUMERICAL PROBLEMS ..................................7

2.1 COMPARISON WITH ANALYTICAL AND NUMERICAL APPROACHES ...............7

2.2 MONTE CARLO DOSE CALCULATIONS...............................................................7

2.3 EXAMPLE: AN 8 MEV ELECTRON HITTING THE LINAC TARGET.......................8

3 BASIC ELEMENTS OF A MONTE CARLO CODE FOR DOSE CALCULATIONS 11

3.1 PHYSICS MODELS ..............................................................................................11

3.2 INTERACTION DATA TABLES.............................................................................11

3.3 RANDOM NUMBER GENERATOR ......................................................................12

3.4 GEOMETRY .........................................................................................................12

3.5 MATERIAL COMPOSITION..................................................................................12

3.6 SOURCE DEFINITION..........................................................................................12

3.7 SCORING .............................................................................................................13

3.8 VARIANCE REDUCTION AND APPROXIMATIONS ............................................13

4 A BRIEF HISTORY ...............................................................................................14

4.1 GENERAL PURPOSE CODES.............................................................................15

5 GENERAL PURPOSE MONTE CARLO CODES IN RADIOTHERAPY.................17

5.1 EGS ......................................................................................................................17

5.2 MCNP ...................................................................................................................19

5.3 PENELOPE...........................................................................................................20

5.4 GEANT .................................................................................................................21

6 RATIONALE FOR MONTE CARLO TREATMENT PLANNING.............................23

6.1 REQUIREMENTS ON UNCERTAINTY IN TREATMENT PLANNING...................23

6.2 WHY MONTE CARLO TREATMENT PLANNING.................................................24

6.3 PHANTOM EXPERIMENTS..................................................................................25

6.4 COMPARISONS FOR CLINICAL CASES.............................................................28

6.5 CONCLUSIONS....................................................................................................38

v

PART II: FUNDAMENTALS OF MONTE CARLO ................................................................40

7 MODELLING OF PARTICLE TRANSPORT..........................................................41

7.1 PHOTON TRANSPORT........................................................................................41

7.2 ELECTRON TRANSPORT....................................................................................43

7.3 INTERACTION DATA TABLES.............................................................................48

8 GEOMETRY AND MATERIAL SPECIFICATION ..................................................59

8.1 VOLUMES ............................................................................................................59

8.2 VOXELISED PHANTOMS.....................................................................................59

8.3 CONVERSION OF CT NUMBERS INTO TISSUE PARAMETERS .......................59

9 ACCELERATOR MODELLING .............................................................................66

9.1 GENERAL ASPECTS ...........................................................................................66

9.2 MODELLING OF THE LINAC HEAD.....................................................................67

9.3 VIRTUAL SOURCE MODEL .................................................................................68

9.4 BEAM MODIFIERS...............................................................................................71

10 DOSE SCORING ..................................................................................................75

10.1 DOSE DETERMINATION .....................................................................................75

10.2 SCORING GRIDS.................................................................................................76

10.3 SPATIAL RESOLUTION .......................................................................................77

10.4 CONVERSION OF MONTE CARLO RESULTS TO DOSE TO WATER ...............78

11 VARIANCE REDUCTION TECHNIQUES AND APPROXIMATIONS ....................80

11.1 INTRODUCTION...................................................................................................80

11.2 VARIANCE OF A MONTE CARLO CALCULATION..............................................81

11.3 VARIANCE REDUCTION TECHNIQUES .............................................................81

11.4 RISKS OF VARIANCE REDUCTION ....................................................................89

11.5 DENOISING..........................................................................................................91

12 MONTE CARLO TREATMENT PLANNING ..........................................................96

13 4D MONTE CARLO DOSE CALCULATIONS .....................................................101

PART III: MONTE CARLO TREATMENT PLANNING IN PRACTICE.................................106

14 MONTE CARLO DOSE CALCULATION ENGINES FOR TREATMENT

PLANNING..............................................................................................................107

14.1 PIONEERING WORK .........................................................................................107

14.2 DPM....................................................................................................................109

14.3 MCDOSE/ MCSIM ..............................................................................................110

14.4 VMC, XVMC, VMC++..........................................................................................110

vi

14.5 PEREGRINE.......................................................................................................112

14.6 MACRO MONTE CARLO (MMC)........................................................................113

14.7 DOSE ENGINES SERVING AS COMMISSIONING TOOL .................................114

15 AVAILABLE COMMERCIAL MCTP SYSTEMS...................................................116

16 MONTE CARLO SPECIFIC ISSUES OF COMMISSIONING ..............................118

16.1 INTRODUCTION.................................................................................................118

16.2 PARTICLE SOURCE AND BEAM MODIFIERS ..................................................119

16.3 SEGMENTATION ...............................................................................................120

16.4 NORMALIZATION / MU DETERMINATION........................................................120

16.5 VARIANCE REDUCTION....................................................................................121

16.6 LITERATURE DATA ON MCTP VERIFICATION ................................................121

16.7 CONCLUSION....................................................................................................126

17 RECOMMENDATIONS.......................................................................................127

17.1 COMPARISON OF DIFFERENT DOSE ENGINES.............................................127

17.2 COMMISSIONING ..............................................................................................128

17.3 CT CONVERSION ..............................................................................................129

17.4 CONVERSION OF DOSE TO MEDIUM TO DOSE TO WATER .........................129

17.5 VARIANCE REDUCTION TECHNIQUES AND APPROXIMATIONS ..................129

17.6 DENOISING........................................................................................................130

18 CONCLUSION....................................................................................................131

REFERENCES...................................................................................................................133

APPENDICES ....................................................................................................................159

APPENDIX A. AN EXAMPLE TO ILLUSTRATE DIFFERENCES BETWEEN THE

MONTE CARLO TECHNIQUE AND ANALYTICAL AND NUMERICAL APPROACHES.....160

A.1 ANALYTICAL TECHNIQUE ................................................................................160

A.2 NUMERICAL TECHNIQUE .................................................................................161

A.3 MONTE CARLO TECHNIQUE............................................................................162

A.4 SUMMARY..........................................................................................................164

APPENDIX B: RANDOM NUMBERS IN MONTE CARLO ..................................................165

B.1 RANDOM NUMBERS IN COMPUTERS .............................................................165

B.2 RANDOM NUMBER GENERATORS..................................................................166

1

Summary

The accuracy of dose calculation engines used for treatment planning in radiotherapy

has increased steadily, ranging from calculations based on measurements, to pencil

beam algorithms and superposition/convolution algorithms. Currently, Monte Carlo

dose calculation engines are implemented in commercial treatment planning software

as it is believed that the Monte Carlo method can provide an accuracy within 2-3 %. It

is important that clinical physicists have insight in these systems, when introducing

them into the clinic. This report tackles this acute problem by providing extensive

information on:

general purpose Monte Carlo codes for photon and electron dosimetry

applications

modelling of particle transport

cross sections

MCTP (Monte Carlo Treatment Planning) specific issues such as linac

modelling, CT conversion, variance reduction techniques, scoring grids

Recent developments such as 4D applications and MCTP optimisation

An important question is whether the added value of MCTP is clinically relevant.

To answer this question an extensive overview of the literature is provided. The main

conclusion is that the MC method has important added value when compared to

pencil beam algorithms. More information is needed when comparing MC to

superposition/convolution algorithms, although the first experiments (comparing

accurate Monte Carlo dose calculation engines to superposition/convolution

algorithms) demonstrate that the MC method will become very important in clinical

treatment planning.

As the Monte Carlo method is, by its nature, very time consuming, a number of

approximations have been included in commercial Monte Carlo dose calculation

engines for treatment planning. This leads to a reduction in calculation time of several

2

orders of magnitude. The impact on the dosimetrical accuracy however is not well

known yet. This report provides an overview of existing Monte Carlo dose calculation

engines, focussing on applied approximations. An overview of commercial MCTP

systems that are already available or are currently being developed is given. As

benchmarking remains as important as for any other treatment planning system, a

paragraph is devoted to quality control. Commercial MCTP systems can be

benchmarked by measurements but also by comparison with accurate Monte Carlo

dose calculation engines containing only a few approximations.

3

Abbreviations

3D Three-Dimensional 4D Four-Dimensional AAPM American Association of Physicists in Medicine ASCII American Standard Code for Information Interchange BEAM an EGS4/PRESTA or EGSnrc/PRESTAII Monte Carlo user code CERN European Organization for Nuclear Research CSDA Continuous Slowing Down Approximation CPU Central Processing Unit CT Computed Tomography CTV Clinical Target Volume DOSXYZ an EGS4/PRESTA Monte Carlo user code DPM Dose Planning Method (MC algorithm for photons and electrons) DVH Dose-Volume Histogram EGS Electron Gamma Shower (a Monte Carlo code) ENIAC Electronic Numerical Integrator And Computer EPID Electronic Portal Imaging Device EPL Equivalent Path Length ESTRO European Society for Therapeutic Radiology and Oncology ETRAN Electron TRANsport (a Monte Carlo code) FORTRAN FORmula TRANslation (programming language) FWHM Full Width at Half Maximum GEANT GEometry ANd Tracking (a Monte Carlo code) ICRU International Commission on Radiation Units and Measurements IMRT Intensity-Modulated Radiation Therapy ITS Integrated Tiger Series (a Monte Carlo code package) KEK National Laboratory for High Energy Physics (Japan) LANL Los Alamos National Laboratory MC Monte Carlo MCDOSE an EGS4/PRESTA Monte Carlo user code MCNP3 Monte Carlo Neutron Photon (a Monte Carlo code) MCNP4 Monte Carlo N-Particle (a Monte Carlo code) MCTP Monte Carlo Treatment Planning MLC Multi-Leaf Collimator MMC Macro Monte Carlo (MC algorithm for electrons) MORTRAN Fortran pre-processor (used for EGS) MRI Magnetic Resonance Imaging MU Monitor Unit NIST National Institute of Standards and Technology NCS Netherlands Commission on Radiation Dosimetry NRC National Research Council of Canada NTCP Normal Tissue Complication Probability PB Pencil Beam PC Personal Computer PENELOPE PENetration and Energy LOss of Positron and Electrons (MC code)

4

PET Positron Emission Tomography PRESTA Parameter Reduced Electron Stepping Algorithm PTV Planning Target Volume RBE RadioBiological Effectiveness QA Quality Assurance SLAC Stanford Linear Accelerator Center SPECT Single Photon Emission Computed Tomography TPS Treatment Planning System TRUS TransRectal UltraSound TCP Tumor Control Probability VISED Visual Editor (graphical interface for MCNP) VMC Voxel Monte Carlo (MC algorithm for electrons) VMC++ MC algorithm based on VMC and XVMC XVMC MC algorithm for photons based on VMC

5

1 Introduction

In the past decades, the sophistication of dose calculation models implemented

in clinical radiotherapy treatment planning systems has gradually improved, together

with available computing power in hospitals. This evolution, going from rather simple

scatter- and inhomogeneity corrections to pencil beams and

superposition/convolution models has resulted in continuous improvements in the

accuracy of predicted patient doses. In superposition/convolution models, pre-

determined Monte Carlo results are used. Full Monte Carlo dose calculations would

therefore seem the next logical step.

For many years it has been realised that full Monte Carlo simulations of the

radiotherapy dose delivery process should further improve calculation accuracy. Due

to limitations in computing power, however, this was never a realistic option in a

clinical setting. Recently, vendors of clinical treatment planning systems have

nevertheless started to offer Monte Carlo dose calculations. However, available

computing power may still not allow for full Monte Carlo simulations in clinical

practice. Approximations and simplifications to speed up the calculations may

therefore be necessary, possibly (partially) jeopardising the advantages of full Monte

Carlo dose calculations.

The aim of this NCS report is to provide potential users of a clinical treatment

planning system with an introduction in the Monte Carlo technique. Apart from

providing an explanation of fundamental and practical aspects specific to Monte

Carlo treatment planning, recommendations (although limited) for potential users and

vendors are included. This report only covers external photon and electron beam

therapy using conventional linear accelerators. Brachytherapy, hadron therapy,

tomotherapy, robotic radiotherapy, etc., are beyond the scope of this report.

6

Part I: Introduction to Monte Carlo

7

2 Monte Carlo for solving numerical problems

2.1 Comparison with analytical and numerical approaches

The main difference between the Monte Carlo technique on one hand and

analytical and numerical approaches on the other is the use of a random number

generator and a set of probability distributions to sample parameter values for

calculating a possible solution to the problem for a single case or event. By

simulating many cases or events, reliable average values can be obtained. Since

the result is an average, it is associated with a standard deviation that expresses the

uncertainty due to the fact that the simulated number of events is less than infinite.

This source of uncertainty is not present when analytical methods are used. Of

course, the answer obtained with analytical methods is still associated with an

uncertainty, arising from the common sources such as uncertainties in the input

parameters and possible systematic errors in the model. A possible disadvantage of

analytical methods is that solutions may be difficult to obtain for complex problems.

(Minor) changes in the relationship between parameters, or the introduction of a new

parameter, may create a major problem in finding a new analytical solution.

Numerical methods are generally less sensitive to such changes. If, for

instance, a relationship changes, the numerical algorithm can stay the same,

because it only uses the values of the function at certain points. In Appendix A, the

example of calculating the area of a circle with radius 1 is used to demonstrate some

differences between the different techniques.

2.2 Monte Carlo dose calculations

In a Monte Carlo dose calculation, the track of each individual ionizing particle

(in radiotherapy generally photons and electrons) through the volume of interest is

simulated. Along its way, the particle may interact with the matter through which it is

passing, e.g. through Compton scattering (for photons) or Coulomb scattering (for

electrons). Using a random number generator and probability distributions for the

different types of interaction, the program samples the distance l to the next

interaction for a particle at a given position and with velocity vector v in a certain

8

direction. The particle is then propagated with velocity v over the distance l to the

interaction location. Next, the program chooses the type of interaction that will take

place. For a dose calculation, one extra step is needed. The dose is defined as the

amount of energy deposited per unit of mass (J/kg = Gy in SI units). Therefore, for

each interaction that is simulated, the program calculates the energy balance: the

energy of the incoming particle(s) minus the energy of the outgoing one(s). To

calculate the dose in a particular volume (voxel), one adds the contributions from all

interactions taking place inside the volume, and divides this by the mass in the

volume.

2.3 Example: An 8 MeV electron hitting the linac target

To illustrate some of the principles of Monte Carlo dose calculations, the

simulation of a photon that is generated in a linac head when an 8 MeV electron hits

the target is described. The energy distribution of the photons generated is depicted

in the left panel of Figure 2.1 The photon energy can be determined in two ways. The

first one is the so-called hit-or-miss method. For this method, two random numbers

are generated, one of which, designated x, is uniformly distributed between 0.01

and 8 (photon energy), the other, y is uniformly distributed between 0 and 1.2

(probability density of a photon with that energy). The value of 1.2 is chosen to be

equal to the maximum of the energy probability distribution (left panel of Figure 2.1),

or slightly above that. The point x,y is now plotted in this probability distribution. If it

is above the curve the target was missed, the point is rejected, and a next point is

randomly generated. If it is below, the point is accepted, and the photon energy is x

MeV.

9

Figure 2.1 Left panel: energy probability distribution for photons that are

generated when an 8 MeV electron hits a linac target. Right panel: cumulative

probability distribution generated from the left panel. The cumulative probability at a

certain energy is the probability to generate a photon at or below that energy.

At first glance, it may seem that there is a reasonable chance that a chosen

point x,y will end up below the curve, yielding the hit-or-miss method rather efficient.

However, the probability density in Figure 2.1 is plotted on a log-scale. Therefore, a

large number of points will be rejected.

A more efficient method for selecting photon energies is based on the

cumulative probability distribution (right panel of Figure 2.1). For this method, values

for the cumulative probability are randomly selected, using a single random number,

uniformly distributed between 0 and 1. Figure 2.1 shows an example for a selected

value of 0.732. The corresponding energy, in this case 1.3 MeV, is selected. This

algorithm is very efficient because only one random number is needed, and each

value results in the selection of a photon energy, i.e. there is never a miss.

Apart from the photon energy, the angles and between the directions of the

incoming electron and the created photon have to be selected. Also for these angles,

probability distributions are known. Therefore, the Monte Carlo program can generate

values for and in exactly the same way as for the energy. Once the energy and

angles of the photon are known, the distance to the first interaction site can be

10

selected, using the attenuation coefficient ([m-1]), which is the product of the atomic

cross sections ([m2]) of the materials that the photon encounters, and the atom

density of these materials ([m-3]). The probability that the photon will travel a distance

l without undergoing any interactions is then given by exp(- l ), and d l is the

probability to interact in the interval d l . So, the probability for an interaction between

l and l +d l is given by exp(- l )d l . Similar as for the selection of the photon

energy (Figure 2.1), a cumulative probability curve P( l ) can now be constructed for

selection of the (first) interaction site:

(1.1)

From this cumulative probability distribution of distances, the travel length l for

a random number r in the range [0,1] can now be expressed analytically:

(1.2)

Here, 1-r is again a random number that is uniformly distributed between 0 and

1; in the final step it has been replaced by a new random number, r.

With the travel distance l to the first interaction site known, the position of the

photon can be updated, and the type of interaction that will take place can be

selected, based on the cross section data for the different interactions. Subsequently,

the energies and angles of the particles that are produced in the interaction are

generated, and the whole process is repeated until all particle energies are below a

pre-defined cut-off energy.

===

l ls edselP0

1)( L

)'ln(1

)1ln(1

1 rrler l

===

11

3 Basic elements of a Monte Carlo code for dose

calculations

3.1 Physics models

The physics models are usually hard-coded in the Monte Carlo software.

Photons are transported in a way that is analogue to reality. For electrons, the

simulation of each individual interaction is very time consuming and impractical for

radiotherapy applications. Therefore, so-called condensed history techniques have

been introduced (section 7.2). These techniques are approximations of the real

physics, and implementation differences exist between different codes. This may

lead to different results, which is the main reason why these codes need to be

thoroughly benchmarked. Even with condensed history techniques, electron transport

often remains the most time-consuming part of radiotherapy Monte Carlo simulations.

The user may be able to manipulate the physics modelling via a number of so-

called transport parameters. For example, the user may enable/disable certain

interactions and/or set the values of parameters that determine e.g. cut-off energies

or electron step lengths. Such parameters may significantly influence a simulation.

For example, when a particles energy decreases below the cut-off energy, it is

discarded and the remaining energy is deposited locally. Obviously, increasing this

parameter will increase the calculation speed, but accuracy might be lost. See

sections 7.1 and 7.2 for details.

3.2 Interaction data tables

Data tables with interaction probabilities for each type of interaction for each

element are usually provided together with a Monte Carlo program. Each of the

Monte Carlo programs has its own format for these tables, therefore interchanging

data tables between the various Monte Carlo programs is a non-trivial task. However,

since these data tables are so closely linked to the Monte Carlo program, the

installation of the program will typically also include installation of the data tables (see

section 7.3).

12

3.3 Random number generator

By its nature, the Monte Carlo method requires a random number generator for

sampling the probability distributions. In computer codes, this is generally solved by

implementing a recurrence relation. Properties such as uniformity of distribution and

random number sequence length are crucial for the reliability of the Monte Carlo

code. This topic is addressed in more detail in Appendix B.

3.4 Geometry

The geometry is to be specified by the user. Depending on the code, different

geometric structures can be defined: planes, cylinders, spheres, cones, and

sometimes even more complicated structures, see section 8.1. In some general

purpose Monte Carlo codes, an (additional) scoring geometry has to be introduced in

regions where the dose distribution is to be calculated.

3.5 Material composition

All materials present in a simulation must be specified by the user. In most

programs, the materials are specified in terms of their elemental composition and

density (see chapter 8). Sometimes additional information is required to enhance the

accuracy of modelling.

3.6 Source definition

The tracking of particles starts at a position (or range of positions) where the

energy and angular distributions of the particles are known with some confidence.

For instance, in a linac the energy and angular distributions of electrons hitting the

target are fairly well known. Accelerator modelling is described in more detail in

chapter 9.

13

3.7 Scoring

To extract the absorbed dose distribution from the particle transport simulation,

one has to define a so-called tally or scoring function. More details on this topic are

provided in chapter 10.

3.8 Variance reduction and approximations

To increase the efficiency of Monte Carlo calculations, approximations and

variance reduction techniques have been introduced. Examples of approximations

are the already mentioned condensed history technique for electron transport, and

the use of cut-off energies. Variance reduction techniques are statistical methods that

enhance the efficiency of a calculation. Theoretically, these techniques result in

identical expectation values as without variance reduction, whilst the calculation

speed is increased. In practice, however, care should be taken and each of these

techniques should be benchmarked. More details are given in chapter 11.

14

4 A brief history

The technique of random sampling to solve mathematical problems is quite old.

One of the earliest documentations is by Compte de Buffon in 1770. In the early

nineteen-thirties, using a mechanical adding machine, Fermi already applied

statistical sampling techniques for radiation transport calculations related to neutron

diffusion (Metropolis 1987, Wood 1986). The statistical techniques were, however,

considered impractical as they were time-consuming and tedious. During the second

world war Mauchly and colleagues developed the first electronic digital computer

named ENIAC, Electronic Numerical Integrator And Computer, containing around

18.000 double triode vacuum tubes in a system with half a million solder joints

(Cooper 1989). Development of the ENIAC was inspired by the labor- and time-

intensive ballistic computations for generation of firing-tables. The system was

realised in late 1946, and in 1947 it was moved to its permanent home at the

Ballistics Research Laboratory in Maryland, USA. Very soon it was realised that the

ENIAC offered new opportunities for statistical sampling techniques. The first tests

were on a variety of problems in neutron transport. One of the collaborators, N.

Metropolis, named the mathematical method Monte Carlo, after the city with its

famous casinos (Metropolis 1987, Cooper 1989).

As computers gained speed and memory, the Monte Carlo codes became more

sophisticated. The first version was written in machine code, but by the early 1960s

programming languages such as FORTRAN (FORmula TRANslation released in

1957 by IBM -International Business Machines- and standardised in 1966, 1977 and

1990) got into use. The fast developments in computer hardware and software and in

statistics were of great influence on the application of Monte Carlo techniques. These

Monte Carlo methods on the other hand helped to improve the hard- and software,

and became one of the most important tools of the statisticians.

At first, the development of dedicated coupled photon electron transport codes

for each specific problem required a lot of effort. Today, this is no longer necessary

due to the availability of general purpose codes, like ETRAN, ITS, MCNP, EGS,

GEANT, and PENELOPE. Most Monte Carlo systems dedicated to radiotherapy are

(partially) based on these codes. Therefore, a short history of the most important

15

general purpose codes is given in the following section. The introduction of Monte

Carlo into radiotherapy treatment planning is discussed in detail in section 14.1.

4.1 General purpose codes

The ETRAN (Electron TRANsport) code, developed and maintained at the

National Institute of Standards and Technology (NIST), Gaithersburg, Maryland,

USA, contains the basic algorithms for simulating the tracks of electrons and photons

travelling through matter (Seltzer 1988). The code was originally developed as a tool

for solving electron transport problems involving energies up to a few MeV. Later, the

production and propagation of secondary bremsstrahlung was added, to extend the

calculation to higher energies. The methods used to generate electron trajectories go

back to a paper of Berger (1963), describing the sampling from multiple-scattering

distributions. In the early 1970's, at Sandia National Laboratories, the ETRAN code

was made more user friendly, especially regarding the specification of the problem

geometry, and extensions were made to lower energies by including more elaborate

ionization and relaxation models. The combined software was designated the

Integrated TIGER Series (ITS) system (Halbleib et al 1988). The Los Alamos

National Laboratory (LANL) integrated the electron transport algorithms of ITS 3.0

into their MCNP3 (Monte Carlo Neutron Photon) code, yielding the MCNP4 (Monte

Carlo N-Particle) system, which was first released in 1990 (Briesmeister 2000).

Based on this code, a different group at LANL developed MCNPX, which can be

used to simulate many additional types of particle (Waters 2002).

During the early 1960's, Nagel wrote his Ph.D. thesis at the Rheinischen

Friedrich-Wilhelms-Universitt in Bonn on electron-photon Monte Carlo. The in-house

developed Fortran code was a very practical (freeware) tool for experimental

physicists during the mid 1960's. Electrons and positrons could be simulated from 1

GeV down to 1.5 MeV, and photons were followed down to 0.25 MeV. The code was

limited in geometry handling. From 1972 to 1978, Ford and Nelson from Stanford

Linear Accelerator Center (SLAC) collaborated to revamp Nagels program and make

it more user friendly. In addition, special attention was given to allow for easy future

enhancements. The resulting EGS3 code (Electron Gamma Shower) was introduced

16

in 1978. Nelson (SLAC) and Hirayama (National Laboratory for High Energy Physics,

KEK) extended the flexibility of EGS in general, and for high energy accelerators in

particular. Rogers and colleagues (National Research Council of Canada, NRC)

extended the code to low energies. These efforts were pooled together in 1985, and

EGS4 was introduced (Nelson et al 1985). In 1990, PRESTA (Parameter Reduced

Electron Stepping Algorithm) was introduced in EGS4 (Bielajew and Rogers 1987). In

2000, Kawrakow and Rogers released the EGSnrc code as the successor to EGS4,

with further improvements in the modelling of electron transport (Kawrakow and

Rogers 2000).

PENELOPE (PENetration and Energy LOss of Positrons and Electrons) was

developed by Universitat de Barcelona and Institut de Tcniques Energtiques,

Universitat Politcnica de Catalunya in Barcelona, Spain, and Universidad Nacional

de Cordoba, Argentina (Salvat et al 2003). It was first released in 1996. PENELOPE

performs Monte Carlo simulation of electron-photon showers in arbitrary materials.

Initially, it was devised to simulate the penetration and energy loss of positrons and

electrons in matter; photons were introduced later. Large efforts were made to make

the simulation of electron transport as accurate as possible, especially in the low

energy region.

The first version of GEANT (GEometry ANd Tracking) was written in 1974 as a

bare framework, which initially emphasised tracking of a few particles per event

through relatively simple detectors. The code was developed as a simulation tool for

high energy physics experiments. From 1993 to 1998, the FORTRAN based

GEANT3 simulation program was entirely redesigned as an object-oriented program

written in C++, designated GEANT4 (Agostinelli et al 2003). This code is a

collaboration of many international research groups under supervision of CERN

(Conseil Europen pour la Recherche Nuclaire / European Organization for Nuclear

Research). It is a very versatile code, useful for many different types of particles over

a wide energy range and capable of handling complex geometries. GEANT4,

includes a low-energy electromagnetic physics package, which makes it useful for

radiotherapy applications. Recently, an implementation of the PENELOPE

electromagnetic physics has also been added to the code.

17

5 General purpose Monte Carlo codes in radiotherapy

At present, four general purpose Monte Carlo systems are in use for

radiotherapy dose calculation. These systems are EGS (Nelson et al 1985,

Kawrakow and Rogers 2000)), MCNP (Briesmeister 2000, Waters 2002),

PENELOPE (Salvat et al 2003), and GEANT (Agostinelli et al 2003).

EGS and PENELOPE simulate the coupled transport of photons and electrons

(and positrons), while other particles such as neutrons or protons are not taken into

account. This has the advantage that during the development of these codes all

attention has been focused on the particles of interest for radiotherapy dose planning.

On the other hand, in high energy photon beams (18 MV and higher) the production

of neutrons and protons in the accelerator head may impact (the biological effect of)

the physical dose distribution in the patient, especially in bone where even alpha

particles have a non-negligible contribution (Chibani and Ma 2003). These particles

can be taken into account in MCNP and GEANT. The latter codes were not

developed specifically for low-energy (radiotherapy) dosimetry, but large efforts have

recently been made to provide reliable low-energy extensions of these systems.

In the next paragraphs, the four systems are described in more detail, focusing

on the mutual differences. In general, it can be said that modelling of photon

transport is quite similar in all four systems in the energy range of radiotherapy

applications, although different cross section data are used. The main differences

occur in the electron transport, which can be dealt with in several ways, having a

large impact on the speed and accuracy of the systems. In the paragraphs below

only a short introduction is given. For more details, the reader is referred to the

corresponding references. An interesting overview has been given by Verhaegen and

Seuntjens (2003).

5.1 EGS

In the past decade, much attention has been paid to the electron transport in

EGS (Electron-Gamma Shower). In 1990, PRESTA (Parameter Reduced Electron

Stepping Algorithm) was introduced in EGS4 (Bielajew and Rogers 1987), and in

18

2000 the EGSnrc code was released by Kawrakow and Rogers as the successor to

EGS4. In EGS4 (Nelson et al 1985), the Molire (1948) multiple scattering theory is

used, which is only valid for small scattering angles. In EGSnrc (Kawrakow and

Rogers 2000, Kawrakow 2000a), an improved multiple scattering theory based on

screened Rutherford elastic scattering is used instead. Furthermore, this code uses

PRESTAII (Bielajew and Kawrakow 1997). The main improvement of PRESTAII

compared to PRESTA is the introduction of a single scattering model of electron

transport, making it possible to reduce the electron step length to very small values

near material boundaries. These improvements are expected to improve the

calculation accuracy of angular deflections for electrons, eliminate restriction on the

maximum and minimum electron path length in EGS4/PRESTA-I imposed by the

Molire theory, and provide an exact boundary-crossing algorithm by using single

elastic collisions of electrons.

From the benchmarks applied to EGSnrc (Kawrakow 2000b, Verhaegen 2002),

it can be concluded that this code is very accurate even in the vicinity of interfaces

between materials with high and low atomic numbers (Z). However, for MCTP

applications EGS4 (PRESTA) seems good enough and is faster than EGSnrc. A

disadvantage of EGS4 and EGSnrc is that users need to program their code in a

macro Fortran code called Mortran. Obviously, only the geometry, source input, and

tallying need to be programmed. In a pre-compilation step, the user code is

connected to the EGS core.

Two user codes, designated BEAM and DOSXYZ (Rogers et al. 1995, Rogers

et al 2002), are available for applications in MCTP. BEAM is an EGS user code

specifically developed for the modelling of a linear accelerator. All components of the

accelerator (target, primary collimator, flattening filter, monitor, jaws, MLC, etc.) are

pre-programmed in so-called component modules. The user can build an accelerator

by simply summing the required components. An input file must be generated in

which the dimensions, materials and transport parameters of the individual

components must be defined. No programming efforts are required. With BEAM it is

possible to determine so-called phase-space files in a plane at the exit of the linear

accelerator. These files contain all necessary parameters (direction, location, energy,

charge, etc.) of particles passing through the plane. Such files can then be used as

19

input for dose calculations in phantoms or patients using the other pre-programmed

user code, designated DOSXYZ. In this code CT data can be imported and translated

to voxels with a certain material and density. Systems as MCDOSE, Peregrine,

XVMC and DPM (section 13) are totally or partially based on BEAM and DOSXYZ.

5.2 MCNP

MCNP is a general-purpose, continuous-energy, generalised-geometry, time-

dependent, coupled neutron/photon/electron Monte Carlo transport code. Two

versions of the MCNP (Monte Carlo N-Particle) code, developed by different groups,

currently exist. MCNP4C (Briesmeister 2000), is able to simulate the (coupled)

transport of neutrons, photons and electrons, whereas MCNPX (Waters 2002) can

simulate a variety of other particles as well. The photon and electron physics in the

present version of MCNPX (version 2.5) are identical to those in MCNP4C. Hence, in

the following we will denote both codes as MCNP. It is noted that the successor of

MCNP4C, MCNP5 (Brown 2003), has been released, but is not yet available outside

the USA.

The electron transport algorithms in MCNP are claimed to be equal to those in

the ITS 3.0 system (Halbleib et al 1988), which in turn were derived from ETRAN

(Seltzer 1988). The Goudsmit-Saunderson multiple scattering theory is used, while

the sampling of energy loss is based on the Landau straggling theory. Several

investigators have shown though that care should be taken with the electron

transport (Jeraj et al 1999, Schaart et al 2002, Reynaert et al 2002). A systematic

error is present in the default MCNP electron energy indexing algorithm. However,

the user can choose to use the ITS electron energy indexing algorithm instead, which

leads to correct results. An additional problem exists with MCNP4C when the

geometry contains many boundaries, e.g. in the case of a voxelised phantom.

MCNP4C requires the voxels in such a phantom to be modelled as separate material

regions, even if they exist of the same material. It has been shown that in such cases

the cumulative effect of many small boundary crossing artefacts may lead to

significant errors in the calculated dose distribution (Schaart et al 2002, Reynaert et

al 2002).

20

In contrast to EGS and GEANT4, MCNP does not require any programming by

the user. Instead, the user only needs to provide an ASCII input file specifying the

problem geometry (using a variety of available surface types and/or macrobodies

such as spheres, boxes and cylinders), the source(s) (energy and angular spectra,

etc.), the tallies (e.g. energy deposition or track length), and (optionally) the use of

one or more of the many available variance reduction techniques. The simulation

results are provided in ASCII output files. Graphical user interfaces, such as VISED

(2004) are available to generate input files and to visualise the output data.

5.3 PENELOPE

PENELOPE (PENetration and Energy LOss of Positrons and Electrons) has

been introduced recently (Sempau et al 1997, Salvat et al 2003). The code simulates

the coupled transport of electrons, positrons and photons with energies between a

few hundred eV and 1 GeV. It is capable of handling complex geometries and static

electromagnetic fields. Large efforts were made to make the simulation of electron

transport as accurate as possible. Ideas introduced in PENELOPE have been

implemented in EGSnrc and vice versa. So it can be expected that these codes will

provide rather similar results. In PENELOPE a mixed scheme of single and multiple

scattering is used, comparable to EGSnrc. The multiple scattering algorithms are

based on the Goudsmit-Saunderson theory. In the PENELOPE implementation of

multiple scattering, the angular deflection and the lateral displacement for each

electron step are accounted for using the so-called random hinge method, which is a

simple and fast method for obtaining an accurate geometric representation of the

electron track. The user has to program the application in Fortran, although several

user codes are available in the system. Benchmarks of PENELOPE against other

codes and experiments have recently been published by Sempau et al (2001),

Sempau et al (2003) and Ye et al (2004). These studies generally show good

agreement with EGS and experiments. The applicability for linac modelling has been

illustrated in Sempau et al (2003).

21

5.4 GEANT

GEANT (GEometry ANd Tracking) was originally developed for high-energy

physics. It can be used for the simulation of many types of particle over a wide

energy range. The current version, GEANT4, includes a low-energy electromagnetic

physics package, which makes it useful for radiotherapy applications (Agostinelli et al

2003). Recently, an implementation of the PENELOPE electromagnetic physics has

also been added to the code. The code can handle complex geometries,

electromagnetic fields, (electronic) detector response, and allows for time-dependent

(4D) modelling of e.g. decaying particles and/or moving objects. A variety of

visualization tools is provided, as well as connectivity to data-analysis software and

computer-aided design (CAD) programs (for geometry input). The user must provide

a set of C++ objects that are built upon the Monte Carlo core of the program in an

object-oriented approach.

Recently, GEANT4 has found use in a variety of medical physics applications

(Barca et al 2003, Archambault et al 2004). Some benchmarks of GEANT4 electron

and photon transport against other Monte Carlo codes and measurements have been

published by Carrier et al (2004) and Rodriques et al (2004). These studies showed

good agreement for photons. Carrier et al reported fair agreement for electrons,

although some non-negligible differences with e.g. EGSnrc (4% for a 10 MeV parallel

beam) were found (see also Torres et al 2004). Recently Poon and Verhaegen

(2005) extensively benchmarked GEANT4 against EGSnrc for radiotherapy

applications. In this paper, a very nice overview of the photon and electron transport

physics modelled in the GEANT code is presented for the 3 different electromagnetic

physics models (standard, low-energy, Penelope). For photon beams depth dose

curves are in good agreement except in the buildup zone. For electron beams

differences are more important. It is also illustrated that results depend highly on

transport parameters as e.g. the electron step size. This is even more clearly

demonstrated in the paper of Poon et al (2005), where a more fundamental study of

the electron transport in GEANT4 is performed. Accurate results can be obtained

after careful selection of transport parameters. In that case the code is an order of

magnitude slower than e.g. EGSnrc. As new releases of GEANT4 are continuously

22

improved with respect to the code, it can be expected that the role of GEANT4 in

medical physics may become more important in the near future.

In this context it is interesting to note that the OpenGATE collaboration has

recently released the first version of GATE, a modular, scripted, GEANT4-based

Monte Carlo code which, in contrast with GEANT4 itself, does not require the user to

be familiar with C++ (Jan et al 2004). Although this code was primarily developed for

nuclear medicine applications (modelling of PET and SPECT scanners), extensions

into other domains such as radiotherapy are currently being developed.

23

6 Rationale for Monte Carlo treatment planning

6.1 Requirements on uncertainty in Treatment Planning

An interesting discussion on uncertainty in treatment planning is provided in

AAPM report No 85 of the AAPM Task Group 65 (Papanikolaou et al 2004). As

stated in this report, due to the steep slope of the TCP-and NTCP-dose relationships,

a dose error of 5 % might lead to a TCP change of 10% to 20%, and to even larger

NTCP changes (see also Fraass et al 2003). Clinical effects are already noticeable

for dose errors of 7 % (Papanikolaou et al 2004). Therefore accurate dose

information is required.

Between the dose prescription to a tumour and the actual dose delivery a large

number of steps are involved. During each step, uncertainties are introduced,

accumulating to an overall uncertainty for the full process of dose delivery. An

overview of the various components of uncertainty is given in Table 1 of AAPM

Report 85. An overall uncertainty of 4.3 % (1) is obtained, which is in

correspondence with the more familiar 5 % (1) obtained in previous work (Mijnheer

et al. 1987, ICRU 1976).

Improving the quality of the dose engine, i.e. reducing the uncertainty in the

dose calculation, will reduce the overall uncertainty in the delivered dose. It should be

noted that the use of an extremely accurate dose engine will not automatically lead to

very low uncertainties in clinical dose delivery as several other factors contribute

significantly to the overall uncertainty. However, in AAPM report 85 it is claimed that

the overall uncertainty in the delivered dose will decrease to 2.5 % (1), leading to a

situation where the accuracy of the dose engine plays an important role. At present, it

is generally believed that the dose calculation should be accurate to within 2% - 3%

(1) (Fraass et al 2003).

24

6.2 Why Monte Carlo Treatment Planning

Monte Carlo dose calculation engines have the potential to meet, or even

perform better than, the 3 % (1) uncertainty requirement, regardless of beam

geometry and patient composition. As for any type of dose engine, however, the

uncertainty for a Monte Carlo dose engine will never be zero due to, for example:

imperfect matching of the Monte Carlo beam to the actual accelerator beam,

uncertainties in the cross section libraries,

the standard deviation due to the limited number of histories simulated,

uncertainties in the conversion of CT data to material composition and density.

The quality of beam matching is very difficult to estimate, but in general it should

be possible to achieve this within 1 % (1) or better (Verhaegen and Seuntjens 2003

and Ma, Jiang 1999). Most authors assume that the uncertainty in cross section

libraries is small enough to be negligible (Fraass et al 2003). The statistical

uncertainty depends on the number of histories. The uncertainty associated with

tissue characterization is difficult to quantify. Instead of using water with different

densities for all tissue types, the real tissue composition must be estimated for the

calculation of cross sections.

Taking all of the above-mentioned uncertainties into account, Monte Carlo

treatment planning is expected to be able to offer an uncertainty in dose calculation

well within 3 % (1) required for accurate radiotherapy. Other advantages are given

by Fraass et al (2003). One advantage over conventional dose engines is that the

uncertainties are independent of the treatment setup. Furthermore, the Monte Carlo

method could lead to an increase in confidence in the obtained dose distributions

(see also Cygler et al 2005). This could lead to the delivery of a higher tumour dose

to avoid recurrence, while having faith in the reported dose to critical organs.

An interesting discussion is provided in a point/counterpoint discussion between

Mohan and Antolak (2001). Arguments against MCTP raised by Antolak include: the

influence of (statistical) noise, the influence of approximations and variance reduction

techniques introduced to limit the calculation time and the limited spatial resolution

(voxel size) often used, again to speed up the calculations. These arguments are

considered of minor importance by Mohan: approximations and variance reduction

25

techniques are illustrated to introduce no bias, the effect of statistical noise is very

limited and resolutions up to 2 or 3 mm can be reached within a few minutes of

calculation time. It is clear, however, that the added value of MCTP compared to

superposition/convolution algorithms should be illustrated by examples. In the

following two paragraphs a literature study of phantom studies and comparisons for

clinical cases is provided.

6.3 Phantom experiments

In the vicinity of low density volumes (lung) and air cavities, Monte Carlo dose

calculations have been reported to be more accurate than conventional techniques

(Mohan et al 1997, Solberg et al 1998, Ma et al 1999, Keall et al 2000, Martens et al

2002, Heath et al 2004, Paelinck et al 2005). Mohan et al (1997) stated that

conventional methods (including superposition/convolution techniques) will give rise

to deviations ranging from 5 % to 10 % in the presence of tissue heterogeneities. The

results of Ma et al (1999) illustrate that MCTP is certainly interesting for electron

beams, as e.g. the FOCUS conventional dose calculation algorithm (pencil beam

algorithm) leads to large deviations (up to 15 %) and isodose line shifts of more than

1 cm (see figure 6.1).

26

Figure 6.1: Isodose line shift between results obtained with the FOCUS pencil

beam algorithm (a) and Monte Carlo calculations (b) (reproduced with kind

permission of AAPM from Ma et al (1999)).

For photon IMRT applications, an added value of the MC method can be found

in head-and-neck treatment and treatment of lung cancer, because of the presence

of tissue inhomogeneities resulting in loss of electronic equilibrium. For IMRT the

best available non-Monte Carlo dose calculation engines are based on the

27

superposition/convolution method (Boyer and Mok 1984, Mackie et al 1985, Ahnesj

1989, Keall and Hoban 1996, Yu et al 1995). Ma et al (1999) obtained large

differences between the FOCUS planning system and a Monte Carlo dose engine for

a phantom containing lung or bone layers, even when the superposition convolution

method of FOCUS was used. An interesting comparison of two

superposition/convolution algorithms and the Monte Carlo method for a lung cavity is

provided by Paelinck et al (2005) (see figure 6.2).

Figure 6.2: Comparison of two superpostion/convolution algorithms and Monte

Carlo calculations for a phantom with a lung insert in a 6 MV beam (reproduced with

kind permission from Paelinck et al (2005)).

The Helax TMS system (Nucletron, Veenendaal, The Netherlands)

systematically underestimates the dose in the lung-equivalent cavity by 6 %, while

28

the Pinnacle algorithm (Philips Medical Systems, Best, the Netherlands)

overestimates the dose behind the cavity by 4 %. Also in the work of Crammer-

Sargison et al (2004), significant deviations in lung equivalent material were obtained

for the CadPlan pencil beam convolution algorithm (Varian Oncology Systems Inc.,

Palo Alto, CA). Arnfield et al (2000) obtained substantial deviations between

measurements and superposition/convolution (Pinnacle) in and around lung-

equivalent material, while the Monte Carlo results are in excellent agreement with the

measurements. These deviations become more important when simulating a small

(4x4 cm) high energy photon beam (18MV). Krieger and Sauer (2005) performed a

comparison between the pencil beam (Helax TMS), superposition/convolution (Helax

TMS) and Monte Carlo methods for a multi-layer phantom consisting of styrofoam (to

simulate the low density of lung) and polystyrene layers for regular beams. In

polystyrene, superposition/convolution and MC were in agreement with the

measurements while the pencil beam algorithm deviated by 12 %. In styrofoam,

however, even the superposition/convolution algorithm deviated by more than 8 %

from measurements and MC results.

6.4 Comparisons for clinical cases

In the examples described above, extreme situations were investigated

consisting of one single beam crossing a large lung/air cavity. It is not straightforward

to extrapolate these findings to clinical practice. Therefore in this paragraph we will

focus on examples of realistic clinical calculations. The results are discussed

chronologically and the focus is on the most recent results as these are obtained with

the most recent (and thus most accurate) versions of the available conventional dose

calculation engines.

Wang et al (1998) developed a patient specific Monte Carlo dose engine that

was evaluated for conformal lung treatment. The method was approximate as only

one medium (water) was defined, although density variations where taken into

account. The dose distributions obtained were compared against a conventional dose

engine based on the equivalent path length (EPL) method. The Monte Carlo results

illustrated that 20 % of the planning target volume (PTV) was underdosed, while the

29

maximum doses in cord and heart (two parameters used in the objective function of

the treatment planning system) were underestimated by the conventional system by

more than 25 %. Deviations were attributed to the approximate modelling of lateral

particle transport in low density regions by the conventional dose calculation engine.

In a follow-up study (Wang et al 2002), the same PB algorithm and MC code were

compared for IMRT treatment of five lung patients and four head-and-neck patients.

For one lung patient, a decrease of 10% in D95 and 6 % in Dmean was obtained, while

for the other patients the PTV coverage decreased with 2-5%. For one of the head-

and-neck patients (a patient with recurrence) D95 differed by 9 %. In lung, differences

in D05 and Dmax of up to 10 % were found. Also in the spinal cord, differences larger

than 5 % were noticed. For all head-and-neck patients, dose differences in the optical

chiasm were below 2 %. An interesting conclusion is that larger effects are observed

for individual fields than for the composite plan.

Figure 6.3: Comparison between Konrad Pencil beam calculations and EGSnrc

for head and neck treatments (reproduced with kind permission from Laub et al

(2000)). DVHs are for the PTV and optical chiasm. Abbreviations used: PB (Pencil

Beam), Veri (EGS4 Verification calculation), IM/MC (intensity modulated/Monte

Carlo). IM/MC is the dose distribution obtained from the Monte Carlo inverse planning

system.

30

Laub et al (2000) obtained large differences between the KonRad Pencil Beam

algorithm with 1D inhomogeneity correction (and accounting for lateral electron

transport) on one hand and EGS4 (Nelson et al 1985) on the other hand for head-

and-neck treatments (see figure 6.3).

The Monte Carlo result in the PTV was systematically lower than the PB result,

although it is not clear whether the MC dose was expressed as dose to water

(presence of large air cavity with corresponding low stopping powers in the PTV can

lead to differences in DVH of PTV, see par 10.4 for a more detailed explanation).

Differences were attributed to the rebuild-up behind the air cavity. According to the

MC results the dose constraint in the chiasm was violated.

Francescon et al (2000) compared the superposition/convolution algorithm of

Pinnacle with the Monte Carlo code BEAM (Rogers et al 1995) for mediastinal and

breast treatments. Deviations were below 2.5 % and thus within 2 standard

deviations of the Monte Carlo calculation. Also for single fields and large

inhomogeneities the differences were negligible. The study was restricted to large

beams. As stated by Ahnesj (1989) larger deviations are expected for smaller fields.

Jeraj et al (2002) illustrated that two types of error are introduced when using an

approximate dose calculation algorithm for inverse treatment planning, namely a

systematic error due to errors in the dose calculations and a convergence error

resulting from the fact that the optimised beam settings obtained by the approximate

dose engine will differ from those obtained with an accurate dose calculation

algorithm. In this study, results obtained by Monte Carlo, superposition/convolution

and pencil beam methods were compared. Systematic errors were below 1% of Dmax

in the tumour and slightly larger outside the PTV for the superposition/convolution

method and around 5% for the pencil beam algorithm. The authors concluded that

pencil beam algorithms should be replaced by superposition/convolution or Monte

Carlo algorithms.

Leal et al (2003) compared the Plato PB algorithm (Nucletron, Veenendaal, The

Netherlands) with the Monte Carlo program BEAM for different clinical cases. As

illustrated in figure 6.4, significant differences were obtained when comparing the

DVHs in the bladder and the rectum.

31

Figure 6.4: Plato PB algorithm (TPS) versus Monte Carlo (MC) for a prostate

treatment (reproduced with kind permission from Leal et al (2003)).

As recently stated by Chetty et al (2005): when comparing Monte Carlo results

with conventional dose calculation engines, it would be interesting to distinguish

between effects related to differences in the beam model and effects related to the

particle transport within the patient geometry. Therefore Chetty et al used two

32

versions of an equivalent path length algorithm, namely a version with an

approximate beam model, and one with an accurate beam model that provides

excellent agreement when comparing calculational results with measurements in a

homogeneous phantom.

Figure 6.5: Comparison of equivalent path length (EPL) algorithm with Monte

Carlo calculations (DPM) illustrating the importance of accurate tuning for a

homogeneous phantom (above) and a heterogeneous phantom (below). The results

depicted with best fit are obtained with the accurate beam model. (reproduced with

kind permission from Chetty et al (2005)).

33

These two models were compared with the DPM (see section 14.2 for a

description of DPM) Monte Carlo dose engine for a homogeneous phantom, a

heterogeneous thorax phantom and a lung patient plan (see figure 6.5).

The importance of an accurate beam model was illustrated by the fact that the

EPL algorithm using the accurate beam model (best-fit results) gave rise to a much

better agreement with the Monte Carlo results for 6 MV in a homogeneous phantom.

For the lung phantom though, the disagreement between the mean lung dose of the

best-fit results and the MC method was 30 % for 15 MV. This illustrates (as stated by

the authors) that especially at high energy (15 MV) the inhomogeneity effects

(transport of secondary electrons in low density regions) may be more significant

than beam model approximations.

Figure 6.6: Comparison of Corvus pencil beam algorithm with MCSIM Monte

Carlo calculations for the prostate (reproduced with kind permission from Yang et al

(2005)).

Yang et al (2005) compared the Corvus finite-size PB algorithm (with and

without inhomogeneity corrections) with Monte Carlo calculations (MCSIM, see

34

section 14.3) for 25 coplanar and 5 non-coplanar IMRT plans for the prostate (see

figure 6.6).

For the coplanar plans the agreement between MCSIM and Corvus was within 3

%. For the non-coplanar plans, differences up to 7 % in Dmean and above 8 % in D98

were obtained in the PTV. Another conclusion was that it was necessary to apply the

EPL heterogeneity corrections in Corvus.

Boudreau et al (2005) compared Corvus (with and without EPL correction) with

the Peregrine Monte Carlo method (see section 14.5) for IMRT head and neck

treatment planning (see figure 6.7 and table 6.1).

Table 6.1: Summary of ratios between Corvus and Peregrine results obtained

by Boudreau et al (2005) (reproduced with kind permission of Boudreau et al (2005)).

35

Figure 6.7: Comparison of Corvus system with Peregrine Monte Carlo

calculations (reproduced with kind permission from Boudreau et al (2005)).

For the brainstem, Peregrine delivered on average a 6% higher Dmean, so for

individual patients even larger differences were obtained. The Peregrine system was

extensively benchmarked against measurements (Heath et al 2004).

Reynaert et al (2005) presented a comparison between two Monte Carlo dose

calculation engines (Peregrine and MCDE) and the Helax TMS

superposition/convolution algorithm for a head-and-neck patient (see figure 6.8).

36

(a)

(c)

Figure 6.8: Comparison between the Monte Carlo dose calculation engines

Peregrine and MCDE. In part (a) the MCDE doses (obtained with the MLCE model

for the Elekta MLC) were systematically multiplied by 1.07, illustrating a dose

difference of 7 % in the optical chiasm. In part (b) (lateral profiles of 2x40 and 40x2

beam segments) the cause of the discrepancies is demonstrated (problem with MLC

model). (reproduced with kind permission from Reynaert et al 2005).

(b)

37

In this work it was demonstrated that Peregrine provided systematic errors in

the DVHs in the optical chiasma, due to a systematic error in the leaf projection. The

superposition/convolution results are in acceptable agreement with MCDE. Only one

patient was studied.

Figure 6.9: Comparison of PB algorithm, superposition/convolution (Helax TMS)

and BEAMnrc/DOSXYZnrc Monte Carlo calculations for a head and neck patient.

(reproduced with kind permission from Seco et al (2005))

38

Seco et al (2005) performed a comparison between a PB, a

superposition/convolution (Helax TMS) and a MC system for a head and neck patient

(see figure 6.9).

Large differences were obtained in the PTV but, as stated by the author, this

was largely caused by the fact that the MC dose in the air cavities was expressed as

dose to medium (see section 10.4). In the critical structures the Monte Carlo DVHs

differed significantly from the PB and superposition/convolution results.

The most direct way to determine the added value of MCTP is to try to link

observed differences in dose maps to clinical outcome. This can be done by e.g.

comparing post-treatment CT scans with (1) possible recurrence within the PTV with

regions of underdosage (as predicted by the MC results) and (2) possible side effects

in critical tissues in regions of overdosage. Data on this topic is still missing. An

interesting paper on this topic was published by De Jaeger et al (2003). Lung cancer

patients, originally planned with an EPL algorithm, were retrospectively recalculated

with a superposition/convolution algorithm, illustrating large differences in the mean

lung dose (up to 20 %). The Lyman model was used to illustrate that these dose

differences can lead to complications in lung tissue.

6.5 Conclusions

Based on single beam phantom experiments it can be concluded that well-

benchmarked MC dose engines clearly outperform both PB and

superposition/convolution algorithms regarding dosimetric precision. Also for realistic

clinical plans the MC codes are superior to PB calculations. More studies are needed

to investigate to what extent the replacement of superposition/convolution algorithms

by MC may result in a benefit for clinical plans.

Most published MC results for photon beams were obtained by MC experts with

well benchmarked research systems. The influence of the approximations and

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variance reduction methods introduced in commercial MCTP systems on the

uncertainty are not yet clear. Moreover, even an MCTP system without

approximations or variance reduction methods can contain systematic errors.

Consequently, every individual MCTP system must be benchmarked before clinical

use.

An important conclusion is that Monte Carlo dose calculation engines, when

carefully validated against measurements, provide an additional benchmarking tool

for treatment planning, in situations where measurements are difficult or even

impossible (Mohan 1988).

40

Part II: Fundamentals of Monte Carlo

41

7 Modelling of particle transport

The following discussion will be restricted to coupled photon-electron transport

as this is the focus of the present report. In a recent paper, Chibani and Ma (2003)

investigated the influence of photonuclear reactions in the linac head for high-energy

photon beams (18 MV and higher). The effects of neutrons, protons and alphas on

dose (taking into account the RBE of the particles) are below 0.7 %. Therefore it is

unlikely that these particles will ever be taken into account in a MCTP system for

photon and electron beams.

7.1 Photon transport

In general, the types of photon interaction taken into account in a Monte Carlo

treatment planning code are the photoelectric effect, Compton scattering, Raleigh

scattering and pair production.

In the case of photoelectric absorption, the photon interacts with a (tightly

bound) atomic electron. In this process, which is dominant at low photon energies,

the photon disappears and all of its energy is transferred to the electron, which is

ejected from the atom with a kinetic energy equal to the difference between the initial

photon energy and the electrons binding energy. As a result of this process, one of

the atomic shells is left with a vacancy which is promptly filled by a less tightly bound

electron, resulting in the emission of a fluorescence X-ray or one or more Auger

electrons. In a detailed Monte Carlo simulation, all of the secondary particles (photo-

electrons, X-rays and Auger electrons) may be transported. However, in cases where

this does not significantly influence the end result, computing time may be saved by

switching off the transport of one or more of these types of secondary particles.

In case of Compton scattering, a photon interacts with a free (i.e. unbound)

electron. If the photon energy is high with respect to the binding energy of an electron

in its atom, this electron can be considered free for this purpose. Part of the photon

energy is transferred to the electron. The scattered photon and the electron emerge

from the interaction at angles relative to the direction of the initial photon that are

related to the particle energies because of the conservation of energy and

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momentum. Compton scattering is an important process for the energies of interest in

radiotherapy, especially in low-Z materials.

In Raleigh scattering, essentially no energy is exchanged; only the direction of

the photon is changed, usually by a small angle. Raleigh scattering is also called

coherent scattering since the photon scatters elastically off an entire atom, where all

electrons behave coherently. The importance of Raleigh scattering is relatively small,

but not always negligible.

In the case of pair production, the photon disappears and an electron-positron

pair is created. This process is only possible if the photon energy is higher than twice

the electron rest mass (2 511 keV), and dominates at high energies in dense

materials. The positron created in the interaction will annihilate with an electron when

it comes to rest, resulting in the emission of two 511 keV annihilation photons.

In Section 1, an example is given on how the transport of photons is simulated.

It is explained that, for each photon emitted by the source, the distance to the first

interaction is sampled, based on the probability exp(- l) that the photon will not

interact over a distance l.

The photon is then transported to the location of the first interaction.

Subsequently, the type of interaction to be simulated is sampled, based on the partial

cross sections for the different interactions contained in the interaction data tables

(see section 7.3). The selected type of interaction is then simulated. Here, use is

made of the well-known theories describing the kinematics of the various types of

photon interaction, see e.g. Attix (1986). In case of, for example, a Compton

interaction, the energy and direction for the scattered photon are sampled. If electron

transport is taken into account, the energy and direction of the electron participating

in the interaction are also calculated. This particle is put on the stack for later

transport. Then, the distance to the next interaction is sampled for the Compton

scattered photon, and the process is repeated until the photon is absorbed and all

secondary particles have been transported.

If a photon is transported through a phantom consisting of multiple materials, it

is possible that the sampled distance to the next interaction exceeds the distance to

the nearest material boundary. In such cases, the photon is first transported to the

boundary. Then, the distance to the next interaction is sampled using the cross

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sections of the material into which the photon is entering. The photon track is then

continued into the new material region (without changing the direction of flight).

In some calculations, the transport of the fast electrons created by photons can

be ignored since they transport energy over negligibly small distances and/or

charged-particle equilibrium exists. Since electron transport tends to consume a lot of

computation time in Monte Carlo simulations, it may be attractive to switch off

electron transport in such cases. However, the fast electrons may in turn produce

(bremsstrahlung or X-ray) photons, which may have a significant effect on the end

result. Therefore, some Monte Carlo codes offer the possibility to generate such

secondary photons even if electron transport is turned off. The algorithms used for

this purpose rely on certain assumptions (e.g., in the tick-target approximation it is

assumed that each secondary electron is completely absorbed in the same material

in which the corresponding photon interaction has taken place) and therefore need to

be used with some caution.

The physics of photon transport is implemented very similarly in most modern

Monte Carlo codes. Small details can nevertheless be different, e.g. the handling of

the Compton effect regarding the binding of the atomic electron. A condensed

overview of these differences can be found in Verhaegen and Seuntjens (2003).

7.2 Electron transport

The physical processes to be modelled when simulating the transport of

electrons through matter are elastic scattering by (screened) atomic nuclei, inelastic

collisions with atomic electrons causing either excitation or ionisation,

Bremsstrahlung production, and the emission of X-rays and Auger electrons following

electron-impact ionisation. Nuclear processes (which only occur at high electron

energies) are often neglected. Positrons are sometimes simply modelled as electrons

with the addition that annihilation photons are created when the particle comes to

rest. More elaborate models use separate positron cross-section tables and include

rare positron decay processes such as in-flight annihilation and three-photon

annihilation.

An important difference between modelling of electrons and photons lies in the

fact that photons undergo a relatively small number of discrete interactions per

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particle track, whereas electrons undergo a very large number of Coulomb

interactions with the electrons and atomic nuclei in the material. It is computationally

very expensive to simulate each of these individual Coulomb interactions, and

therefore this is not normally done in general-purpose codes or dose engines for

treatment planning.

Instead, a so-called condensed-history approach is usually applied (Berger

1963). In such a model, each electron track is subdivided into a series of short track

segments, usually called steps. Instead of modelling the individual elastic and

inelastic collisions along each step, the resulting (cumulative) energy loss and

angular deflection are sampled once per step only.

The sampling of angular deflection may be based on a so-called multiple-

scattering formalism. One example is the implementation, in EGS4, of the theory by

Molire (1948). The Molire distribution is a universal function of a scaled angular

variable, which makes it relatively easy to sample the angular deflection for arbitrary

step lengths during a run. A disadvantage of this theory is that it is based on a small-

angle approximation, so large-angle deflections are modelled less accurately.

Another multiple-scattering theory, the Goudsmit-Saunderson (1940) formalism, is

valid for all scattering angles. However, sampling the angular deflection for arbitrary

step lengths during a run is less straightforward, so codes based on this theory (such

as ETRAN, ITS and MCNP) usually sample the deflection angle from stored multiple-

scattering distributions that have been calculated for a pre-selected set of path

lengths during the initiation phase of the run (Berger and Wang 1988).

The sampling of electron energy loss may be done in different ways. A

distinction is commonly made between so-called class I and class II algorithms

(Berger 1963, Rogers and Bielajew 1988), see Figure 7.1.

In a class I code the primary electron is not directly influenced by the generation

of a secondary electron. Instead, energy straggling (i.e., the fluctuation in electron

energy due to differences in the energy lost by different electrons of equal initial

energy traversing the same path length) due to the creation of secondary electrons is

taken into account explicitly in the algorithm used to sample the energy loss for each

electron step. Examples of such codes are ETRAN and MCNP, in which the energy

loss is sampled from the Landau (1944) straggling distribution. An advantage of this

45

approach is that energy straggling is always modelled accurately, even if a high

energy threshold for knock-on production is applied. This may greatly speed up a

simulation if the transport of low-energy secondary electrons is not important. A

disadvantage of the class I approach is the possibility for negative energy loss events

in small voxels. Such events may occur if the energy carried out of a voxel by a

secondary electron created within it is larger than the amount of energy deposited in

the voxel by the primary (and secondary) electron.

Figure 7.1 Different ways to perform a sampling of electron energy loss, class I